B. Kolev - Théorie du second gradient dans le cadre relativiste - Covariance générale versus Objectivité

• The idea is to describe the relativistic hyperelasticity with the =variational relativity of Souriau=.
• Consider a Lagrangian depending on the ==$k$-jet== of variables considered, noted $j_m^k$ ($k$-jet on variable $m$).
• General covariance: every Lagrangian is invariant by any diffeomorphism (not invariance by change of coordinates which is not the same definition).
• Perfect matter by Souriau: a function from the universe the the Euclidean space ${\mathbb{R}}^3$.
• Define the body.
• Introduce a reference in the universe, called the material frame ${\mathcal{F}}^{mat}$.
• What is the conformation ? It is the push forward of the inverse of the metric. It is an invariant of general relativity $K({\phi }^{\ast }g,{\phi }^{\ast }p)=K(g,p)\circ \phi$ .
• This idea of construction of invariants of general relativity, we have ;
  • for any tensor define on the universe, we construct its contravariant version
  • define its definition with respect to the material frame
  • show that it is an invariant of general relativity.
• This theory can lead to a definition of the Classical theory of gradient fluids Introduction of an observer
• In mechanics, we need to define time and space, we define a function from the universe to the real numbers, defining a folliation (a space time structure, or structure 3+1).

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• The key idea is the decomposition of the time as a direction of the flux of matter and the direction of displacement of the observer. We have to frames, one linked to the matter, the other to the observer. If observer and matter are the same, then we are not in relativity.
• Killing vector Limit problem
• need to define a Galilean structure on the universe, a pair of contravariant symmetric tensors (spatial co-metric), and a 1-form which is the clock.
• The conformation tends to the inverse of the right Cauchy Green tensor, the relativistic material acceleration tends to the eulerian acceleration.
• In this theory, we start from a Lagrangian which is covariant general. At the limit, we observe that some invariants are objectives, and some not.

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R. Desmorat - Sur la formulation covariante générale de l’hyperélasticité relativiste